CONCRETE LIBRARY OF JSCE NO. 25, JUNE 1995
EVALUATION OF SHEAR STRENGTH OF RC BEAM SECTION BASED ON
EXTENDED MODIFIED COMPRESSION FIELD THEORY
(Translation from Proceedings of JSCE, No.490/V-23, 1994)
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An analytical method to evaluate the diagonal shear failure strength of RC beams was developed,
in which the modified compression field theory was extended. The effects on shear strength of
effective depth, longitudinal reinforcement ratio, stiffness of longitudinal reinforcement, and
concrete strength were investigated analytically. Using a proposed tension softening relation, the
size effect on the shear strength can be evaluated satisfactorily by this analysis method.
Keywords: modified compression field theory, shear failure, tension stzfiening eject, size eflect
Hikaru Nakamura is an associate professor in the Department of Civil and Environmental
Engineering at Yamanashi University, Kofu, Japan. He received his Doctor of Engineering Degree
from Nagoya University in 1992. His research interests include failure mechanism of RC
structures and application of finite element analysis. He is a member of JSCE and JCI.
Takeshi Higai is a professor in the Department of Civil and Environmental Engineering at
Yamanashi University, Kofu, Japan. He received his Doctor of Engineering Degree in 1972 from
University of Tokyo. His research interests relate to the shear failure of reinforced concrete
structures, dynamic behavior of concrete structures and maintenance and strengthening existing
concrete structures. He is a member of JSCE and JCI.
1.
INTRODUCTION
The shear failure of reinforced concrete structuresis a complicatedphenomenoninfluencedby
many factors. It appears that a full understandingof the mechanismof shear failure is extremely
difficult. Given the difficulties,studies of shear have been aimed at obtaining load carrying
capacity, and many studies have been based on empirical methods using many experimental
results or based on limit analysisusing a simpleassumptionof the failure mechanism.
Recently, concrete structures have become larger and more complex, and newly developed
materials such as fiber reinforced plastic(FRP) have been used in concrete structures. In
attemptingto predict the shearstrengthof such structures,evaluationsbased on empiricalmethods
are not necessarilyeffective.For example, the shear strength of RC beams has been found to
graduallyfall as the beam depth increases.However,it is difrlCultto accuratelyestimatethe shear
strength of such large RC structures because of limitations for experimental conditions.
Furthermore,when new materialswith materialpropertiesquite differentfrom steel, such as FRP,
are applied,the effect of these differentpropertiesmust be evaluatedon the basis of numerous
experimentalresults.Thus the equationto accountfor them must be formulatedagain.
On the contrary, the study to evaluate the shear strength of RC beam analyticallyare recently
performedactivity. One such studyis the modifiedcompressionfield theory proposedby Collins
et a1.[1].Theirtheorygives an analyticalmethodfor predictingthe shear responseof RC elements
satisfyingconditionsof compatibilityand equilibrium,in which crackedconcreteis treated as a
new material with its own stress-straincharacteristics.The method is applicableto reinforced
concrete elements in plane stress conditions and under uniform deformation. Therefore, the
methoddoes not apply to the analysisof RC beams subjectedto shear,momentand axial loading.
The purpose of this paper is to analytically evaluate the shear strength of RC beam section
without stirrups using an analytical method in which the modified compressionfield theory is
extendedto cover analysisof RC beams loadedwith combinedshear,momentand axial loading.
Firstly, the analyticalmethodis verifiedby comparingwith an empiricalmethod.The effects on
shear strength of several factors are then investigatedanalytically.In particular, the effect of
stiffness of the longitudinalreinforcementand the size effect are evaluatedanalytically in this
Paper.
2. ANALYTICAL METHOD BASED ON EXTENDED MODUIED COMPRESSION FIELD
TfaORY
The modified compression field theory is a new analytical method for reinforced concrete
elements subjected to shear developed by Vecchio and Collins[1]. The theory, however, is
applicable only to elements subjected to uniform shear, and cannot be applied directly to
reinforcedconcretebeams subjectedto shear and momentloading.
In this paper, an analytical evaluation of the shear strength of RC beams is performed by
extendingthe modifiedcompressionfleld theory.This is achievedby consideringthe cross section
to be composedof a seriesof layers.The conceptof this analyticalmethodis similarto Collins's
method[4].However, the use of a fast computationalgorithmand a tension softeningcurve well
suited the analysisof RC beams differentiateour approach.
W
Consider a uniform cracked reinforced concrete element subjected to uniform shear and axial
loading as shown in Fig.-1. Then considerthe following averagestrains: strain in the longitudinal
direction(ex), strain in the transverse direction(ey) and shear strain(yxy) or the principal
compressive strain(e2), Principal tensile strain(e1) and the angle of inclination of principal
compressivestrain(0). The compatibilityconditionsin terms of average strain in such an element
are written as followsfrom Mohr's strain circlein Fig.-2.
-94
-
y" -2(erex)tanO
(1)
e, -e1 -(el -ex)tan2o
(2)
e2 - e,1e1
(3)
-ex)tan2o
Assume that a shear stress(Txy)and a longitudinal stress(Gx)are acting on the concrete element
and that all of the shear stress is carried in the concrete. Then, the average stress relations are
representedas followsfrom Mohr's stresscircle in Fig.-3.
ox -crl-T"
(4)
/tan0'
cry -Gl-T"tan0'
(5)
o2 - G1 -T"(tanOf+1/tanO')
(6)
in which Gxis the longitudinal stress, oy is the transverse stress, Txyis the shear stress, Gl is the
principaltensile stressand o2 is the principalcompressivestresswhichhas the inclinationof O'.
The inclinationof the principalcompressivestresscoincideswiththat of the principalcompressive
strain, 0-0', if we ignore shear transfer at the concrete crack surface.
For equilibrium in the longitudinaldirection,
N - Asxfsx + oxBwh
(7)
in which N is the applied axial force, Asxis the area of the longitudinalsteel bar, fsxis the stress
of the longitudinalsteel bar, Bw is the width of the concrete element,and h is the height of the
concreteelement.
The equilibrium condition in the transverse direction is as follows, assuming that the transverse
force does not act on the cross section.
oy - -Awfq /BwS
(8)
in which Awis the area of the transversesteel bar, fsyis the stress of the transversesteel bar, and
S is the spacingbetweentransversesteel bars. The stress of the transversesteelbar vanisheswhen
a cross sectionwithout stirrupsis considered.
The equilibriumcondition in shear is
(9)
TRY -V/Bwh
whereV is the shearforce actingon the crosssection.
1
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8
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0 '.T
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l
i Fig:l
ReinfTorced
ConcreteElement
T ;r y
I)
[
L
:2 8
e?
/
/
fey
h
/
/
T
E 'T
l
i
2 8
q y
ey
N
E1
,e
E>
q .,
o '.T
;
0 -1
0-
I
I
l
X
Fig.-2 Mohr's Strain Circle
-95-
Fig:3 Mohr's Stress Circle
The stress and strain satisfyingthe strain compatibilityconditionand the equilibriumof stressesis
obtained when constitutive relations for the concrete in the principal direction and for the
longitudinaland transversesteel bars are assumed,and the stresses obtainedfrom the equilibrium
condition are identical to the stresses obtained from the assumed constitutive relations. This
method, however,requires a trial and error solutiontechnique,since a direct solution cannot be
obtained.
W
Momentand ShearForces
A
n outline of the new analytical method is given here. The stress-strainrelationship for the
concreteand steel bars, then, is alreadyavailablefrom the mechanicalpropertiesof the materials.
0 subdivide the cross section into m layers and give geometrical conditions for each
er(Fig:4(b)).
l8 Give the axial forceP)
and shear force(V)actingon the cross section.Here, it is assumedthat
the shear stress(Txy)is distributedthe uniformlyin the cross section@ig:4(d)).
@ Assume a longitudinal strain. The strain in each of the layers is fixed by defining the
curvature and top strain(ec)in the section, assuming the distributionof longitudinal strain is
linear(Fig. -4(c)).
gxt. - (h-yci)0+ec
(10)
This assumptionof longitudinalstrainand shear stressdistributionis somewhatsimplified.Thus, a
more detailedinvestigationwill be neededin future.
@ satisfy the conditionof compatibilityand equilibriumfor the known Txyand exiin each layer.
The followingprocedureis performedin eachlayer:
(a) Assumethe principaltensile strain(e1).
(b) Calculatethe principaltensile stress(o1)correspondingto el usingthe concretestress-strain
relationship.
(c) Calculate0 from the equilibriumin the transversedirection.
From the transverseequilibrium,the stressof a transversesteel bar is
f9, - -BwS/Aw(o1
-T"
tanO)
(ll)
On the otherhand,the stressof a transversesteelbar obtainedfromthe compatibility
conditionand stress-strainrelationshipof transversesteelis
f.q -ES,ie1 -(e1 -e,)tan2
(12)
ol
Consideringthe condition fsyj'sy using Eq(11) and Eq(12), we obtain a quadraticequationin
tano before the steelbar yields or a linear equationafter yielding.From the algebraic
equationsof tanO,we can identify 0 positively.
(d) Calculatethe principalcompressivestrain(e2),the transversestrain(ey),and the shear strain
(yxy)from the Mohr's strain circle.
(e) calculatethe longitudinalstress(Gx),
transverseStress(Oy),
andthe principalcompressive
stress (G2) from Mohr's stress circle.
(i) Calculate cf'2 from e2 using the
concrete stress-strain relationship.
(g) If o2 does not equal G'2, return
to step(a) andtry anothervalueof
e1.
+
@ calculate the stress of longitudinal
steel bar(fsxi) from the longitudinal
strain of the steelbar.
TJ=y
-.:i:_
I5
+
a
A
e .I:i
P
yet
@Repeat (a) to (g) until the
equilibriumand compatibilityconditions
are satisfied for all layers.
"
B ,u
+
+
(a) cross sccLion
O))collCretelayer (c) distribudonof (a) distribution
!ouna%hdina'
soiesshsear
Fig:4 Beam Secdon using Layered Model
-96-
0 calculate the resultantof the longitudinalstresses.Also checkthe equilibriumof appliedaxial
force.If the equilibriumconditionsare not satisfied,it is necessaryto readjustthe assumedecand
analysisuntil equilibriumis satisfied.
raf
Q9ecataltchu3
Calculatethe moment acting on the sectionfrom the distributionof longitudinalstresses.
Using the above procedure,the moment acting on the section is calculatedfor the appliedaxial
and shear force. To obtain the equilibriumconditionfor the moment,a convergencecalculationof
curvature is needed. The solution algorithm used by Collins et a1. involves the convergence
process of e2 and 0 for each layer, and that of the longitudinalstrain distributionfor the cross
section. On the other hand, proceduredescribedhere has only one convergenceprocess for el in
each layer and ec for the cross section. Therefore, computationalanalysis is fast using few
unknownsto reach convergence.
The ultimate states are defined in the analysis by the following two cases when curvatureis
increasedunderthe conditionof constantaxial and shear force.
(1) The maximumcompressivestrain in the cross sectionis reachedat the strain correspondingto
the strengthof the concrete(-0.002).
(2) No solutionsatisfyingthe equilibriumfor the given exand Txyis obtainedin at least one layer,
evenif the equilibriumconditionfor axial force is satisfied.
Therefore,the ultimateloading capacityis obtainedfrom the moment correspondingto each cases.
The failure modes are thus distinguishedbetweennexural failure for case(1)and shear failure for
case(2). Using the above definition, it is possible to define the ultimate loading capacity
dependingon the differentfailuremode in the analysis.
W
4&
The stress-strainrelationshipfor the concreteused is shownin Fig.-5.In the zone of compression,
the relationship is representedby a second degree parabolaup to the maximum compressive
stress. The maximum compressivestress decreasesdependingon the principaltensile strain el
using an equationproposed by Collins et a1.[3].However,Miyahara et al.[5] reported that the
maximum compressive stress becomes constant value for higher tensile strains in cracked
concrete. Thus, it is assumed in the analysis that the maximum compressivestress becomes
constant when the principal tensile strain exceeds 40 times the crack strain. The stress-strain
relationshipbeyondthe compressivestrengthis not consideredsincethe object of this analysisis
shear failure.
In the tension zone, the stress increases linearly with a constant proportionality of 2f'c/ecoup to
the tensile strength(ft).ARer that, it is evaluated by the following equation which modifies the
equation proposed by Collins et a1.[3].
0 -3
fy
0 -1
0 -i
f c/
I (
I
I
E 3
A
I)
- ey :
I
(
I
l
I
I
(
A
A
e co
Ey
l
(
I
I
I
I
l
I
I
l
l3 F:
[J
c
I I_< C a
A
c c 7.
L-E l 1
Fig:5 Stress-StrainRelation of Concrete
- fy
Fig.-6 Stress-Strain Relation of Reinforcement
-97
-
01
ft
1+cL
200(el -ec,)
(13)
This equation modifies the stress reduction by introducingthe coefficient a into the Collins
equation. The concept of coefficienta and the equation is based on the following discussion.
Equation(13)is the relation between averagetensile stress to averagetensile strain in a cracked
reinforced concrete element. The Collins equation, however,was formulated from experimental
result for RC membraneelementsarrangedwith a reinforcementmesh having equi-gridspacing.
Thus, we predicted that it is not applicable to RC beams with the reinforcement arranged
concentricallyto the lower part of the cross section. Moreover,it is guessed that the bonding
effect between concrete and steel which dominates this relation operates over a specific area
aroundthe steel bar. However,it must be assumedthat the effectinfluencesall tensile area of the
cross sectionin the analysis,becausethe area has not been clarifiedat present.The correctionto
the area innuenced by the bondingeffect is accomplishedby coefrlCientcl. This is the feature in
this studythat the stress-strainrelationshipin tensile,that is tension softeningcurve, is varied by
coefficientcL
b&
The stress-strainrelationshipfor the reinforcementis shownin Fig:6. It is assumedthat the stress
is proportionalto the strain with the initial stiffnessup to the yielding point,and that the yield
stressremainsconstantafter that in bothtensionand compressionzones.
3. INVESTIGATION OF APPLICABnITY
FELD TfEORY
OF EXTENDED MODUIED COMPRESSION
uLBfBQ
As mentioned above,it seems that analysisusing the tension softeningcurve proposedby Collins
et al. overestimatesthe shear strengthofRC beams. We thereforeestimatethe value of a so as to
define a tension softeningcurve suitablefor RC beams.
The model used in this analysisis a cross sectionof20 X 20cm and a beam depth of 16cm as
shown in Fig.-7.The materialpropertiesare that compressivestrengthof concrete is 280kgE!cm2 ,
iegnES!liA
9trZnnF?h
eOfi
n?S:1Crsettieffise
si 8.kf%Cem,e2i
;fE?ceeAieeLfi
Tsg
1?;re;S.0.f
6thkggrceAnf20
TciEeenct,.isss
3s7e8ctO1.n
is subdividedinto 20 layers and the applied axialforce is zero in the analysis.
The relationships between shear strength and shear span ratio(a/d) are illustrated in Fig.-8, in
which cLtakes the values 1,3, and 5. The analyticalresults for cFl are shown with mark Tt+ H,
that of a-3 are shown with mark" A '' and that ofcF5 are shown with mark 'fI ". The marks in
the figure are obtained accordingto the definitionof failure mentioned above. Then, the values of
a/d are calculated supposing a-MrV for two-point loading and a simply supported beam. As a
result, the failure section in the beam is defined in the
maximummoment section.In the figure, the broken line
20cm
indicates the moment capacity curve and the solid line
indicates
the shear strength curve obtained from the
equation proposed by Niwa et al.Pq(14))[6]. The
+
D16
.i_Iig
supposingthat this equationgives the correctsolutionfor
diagonaltension failure strength since it was formulated
from many experimentaldata and its reliabilityis already
proven. It should be noted that the following analysisis
restrictedto RC rectangularsections.
-98
-
a;
U
l?nPvP!1;:iagbiI5
bOyf
tch.empTrias1.ynsis
w?tfhSRs=a,sstreenqe2ti.i:
sE
u
T1
D19
+
Fig.-7 Analytical Model
v - o.94j::1/3(100Pw)u3(d / 100)-u4(o.75.
1.4d / a)Bwd
(14)
in which f'c is the compressivestrength of concrete,Pw is the longitudinalreinforcementratio, d
is the effective depth of the cross section, a/d is the shear span ratio and Bwis the width of the
cross section.
As shownin Fig:8, the results for cF1, which the Collinsequation is used, overestimatethe shear
strength and results with cF5 underestimateit. On the other hand, results with cF3 show good
agreementwith Eq.(14) in the range of diagonal tensionfailure a/d>3.
G1
ft
1+3
200(e1 - ec,)
(15)
Analyticalresults are in good agreementwith the moment capacitycurve in the range of a/d>6.
They are also in good agreementwith the shear failure curve in the range of 3<a/d<6. It is
therefore clear that this analysis can calculateboth shear and flexure failure strengthwhen the
shear span ratio is varied, as long as the tension softeningcurvein Eq.(15)is applied.Further, the
analyticalresults for both ranges correspondto the definitionof the failure mode as mentioned
above. However,the strengthis underestimatedwhen a/d is less than 3. This is the reason for the
shear strengthincreasing as a/d becomes smallerby the effect of support and loading point; that
is, the effect of compressivestress in the transverse direction.However the analysis does not
consider this effect. Consequently,this analysis can evaluatethe nexure and diagonal tension
failure strength, but it will be necessary for the shear compression failure strength to be
investigatedin more detail,
Many tension softening curves have been proposed. The curve proposed here is similar to that
proposedby Okamura and Maekawa[7]for deformedsteel bars, as shown in Fig.-9. Therefore,it
is guessed that the proposed curve is adequate for RC beams. However,further modificationof
the curve for effectivedepth is investigatedin next section, since the averaging bonding effect
between concreteand reinforcementis innuencedby the effectivedepth.
+1
8
E
V7
O
B6
b5
a)
it
A
Li
uA
AA
A
•`
6;4
+cy=1
Acy=3
Icy=5
I
rls
30
U
tth
ill
1
tt
A
I
Eq.(14)
Collins(cy
•`
A
b'1 20
Y
I
1•`----.
hq
A
- 1)
Eq.(15)
3
+
2
7Lp< 1
MomentCapacib,
1
2
3
4
5
6
7
8
9
10
Okamura,Maekawa
0
1l
a/d
Fig.-8 Effect of Tension Softening CUR,e
_!a;
L
C
0
O
V
O
L
L
U
U
h
I•`
0
O
a
0.0.5
U
L•`
O
1
a
o.o2.5
Cn
m
m
0
I_
m
m
O
U)
E
•`
1
CL
O
Q
0
o.o1
0.02
E1
O
O
0-T
Cn
Cn
m
0.00l
C
O
O
O
7.Ty
IC
0
0.0005
Fig.-9 Comparisonof Tension SoftemingCurves
(
O
a
i
lO
50
k9//Cm2
0
-50
-100-150
-200
Fig:lO Distribudon of Stress, Strain, and Angle
-99
-
a
0
de91.ee
20
40
60
:igup;?;
!10,
aslhoc:i;h,ee
ssTvaiytsitcr:i
nre,suusltt
sbOeff.t,hee
fdasltun:eS:Stf?f
S:a5l.nb.ssr,e
sa=a2n.d.
8anXg1
;.ofif?Cii)?
atiohne
principaltensile strain and shear strain increaserapidlynear the lower part of the cross section,
and the longitudinalstress becomescompressiondue to the effectof the bi-axialstress field in the
lower part.
QiBgbgW
It is predicted that the results of analysis will be influenced by the number of subdividedlayers.
Thus, the effect of subdivided layer number is investigated here. The analysis is performed for
four cases, in which the cross section shownin Fig:7 is subdividedinto 40, 20, 10 and 5 layers.
Figure-11 shows the relationshipsbetween shear force and moment obtained in this analysis. The
results for 5, 10, 20 and 40 layers are markedwith
'' 0 f', T'A ", Hu " and " X ", respectively. It
is clear that the results converge when number of layers is more than 20 layers. Therefore, we
adopt 20 layers in the following analysis.
4. EFFECTS ON SmAR FAnURE STRENGTHOF RC BEAM SECTIONWITHOUT
STnuUPS OF VARIABLEFACTORS
We investigatethe effectsof variablefactorson shear strengthin this section.The model used in
this analysis is the RC cross section shown in Fig.-7. The analytical results are verified by
comparison with Niwa's equation, which was formulated on the basis of many experimental
results.
a
The analysis in which effective depth was varied was performed under the condition that the
longitudinal reinforcement ratio is constantPw-0.0269). The shear strength obtained from the
analysis for several effective depths is shown in Fig.-12, when M/(Vd)-3.0. The marks with T.+ "
in the figure are the results obtainedby using the tension softeningcurve of Eq.(15) and the solid
line is the value of Eq.(14). The analytical shear strength increases in proportion with effective
depth increase and the difference from Eq.(14) increases as the effective depth increases. In
general, the shear strength of RC beams is affectedby size and Niwa's equationalso incorporates
this effect. On the contrary, this result shows that the analysis cannot evaluatethe size effect.
Figure-13 shows the distributionof Gx and 0 at failure when the effective depth is 16, 64 and
160cm. The distributionscan be identified for every effective depths. This implies that the size
effect does not appear in the analysis. The reason is that the tension softening curve used is the
same(Eq-(15))in spite of the differenteffective depth.
S 45
C)
g40
0
g7
T66
L1:35
cd
a
B30
B
O
L
u? 25
Le 5
em
L
g4
U)3
2
l
20
a)I
0
A
D
X
15
dE
_=
8
51ayers
101ayers
201ayers
40layers
+ Eq.(15)
A Eq.(16)
- Niwa et al.
10
CET
5
20
l
2
3
4
Moment(tfm)
Fig.-i I Effect of Layer Number
40
60
80
loo
120 140 160
EffectiveDepth(cm)
Fig.-l2 Effect of Tension SoftemingCurveand EffTective
Depth on ShearStrength
-100-
Therefore,we try to estimatea tension softeningcurve involvingthe effect of size by varying the
value of ct in Eq.(13).The value of a will be estimatedin comparisonwith Eq.(14)for the cross
section in which effective depth is less than 160cm,because the applicabilityof Eq.(14) to the
size effect is already appreciatedexperimentallywithin the above effectivedepth. Note that the
tension softeningcurve representsthe bondingeffectaveragedover all the crackedarea.
Figure-14 shows the relationshipbetween a and effective depth obtained in the analysis. The
analytical results are marked with TT
C) Tf,and the solid line is the following equation of
relationshipof cLVersuseffectivedepth obtainedfrom an interpolationof the analyticalresults.
cL- 3(d/16)1/3
(16)
EAq
ieln6S)I?:t.SOERq?Fli3!.
iuhT;eslunlVt?1uvl;Tngg
tthhee
;fq6?(c.t6)Ofa,eefSehC.tivne
bdyeRth^
i.?iZbfqg?_eld2.blyt
iSsu21SebiTt.lhnagt
shows
the size effect can be evaluatedanalyticallyusing Eq.(16) in a similarto q...
(k4e).
aFnialguy;lPc-i.5
results,
the analyticalresults for an effectivedepth of 112cm.The marks with.' i
the broken line is the shear strength curve of Eq.(14), and the solid line is the moment capacity
curve. The analytical resultsare good agreementwith Eq.(14) over a wide range.
The physical meaning of Eq.(16) is not clear since it was obtained only from an analytical
comparisonwith Eq.(14). Moreover,we must verify the applicabilityof the equation to very large
cross sections which is not possible experimentally.Thus the applicabilityto the large-scale cross
sections is investigatedbelow using a energyconsideration.
The shear strength(V1,V2) and the strain softening energy(W1,W2) Obtainedfrom analysis using
Eq.(16) and Eq.(15) for d-16, 64, 160 and 1600cm are shown in Table-1. The strain softening
energy is defined as the principalstrain energy absorbedin the crackedconcretesectionup to
failure. That is,
A
Cr.1.
C
O
O
0
•`
O
O
Cn
O
O
A/
Cn
y)
U)
V)
m
O
L
0
t-
U
U
LAJ
tJ<
O
O
r•`
5
V:
aO
C)
4
ttT>-i
3
Ie
•`
a
O
E3
CLJ
- 3(i)1/3
a 6
#
y-
2
a-16(cm)
a
l
-64(cm)
a-160(cm)
41
kg//,=m,2
20
5VL4oo
deg7.ee
40
60
80
100120140160
EffectiveDepth(cm)
Fig.-14Relationbetween EffecdveDepth and a
Fig:l3 Distribudon of u x and 0 using Eq.(l5)
I
I
1
g
Q>
MomentCapacity
I
P25
O
LL
+
/
L
(t]
a)
a20
l
I
+
2
+I
Eq.(l4)
3 4 5
iJ
I
6
7
8++09 10 ll
C
O
/
O
O
I
Cn
O
O
/
Cn
C4
Ch
I
U)
U)
O
uL
•`
tt
1
•`
O
L•`
U
•`
i
%<
I thtl
tt
i-
O
A
•`
a
aO
i
i
/
C)
/
I
Thhh
a/d
Fig.-15 Analytical result for d-1 12cm
i
/
7 z-
DI
O
C)
l5
a -16(cm)
a-160(crh)
d-160000(cm)
C
O
•`
/
q .1. /.
•`
/
/71
1
/
kgf/cm2
/
Fig:16 Distribution of u x and 0 using Eq.(16)
-101-
I
I degree
w
(17)
- I_h;/2Jeec1,
ol(e1)day
It is understoodthat the strain softeningenergy(W2)used Eq.(15) increasesin proportionwith
effectivedepthincrease.The resultsmeanthat the size effect does not consideredobviouslybased
on the considerationof energy in this case. On the other hand, energy(W1)used in Eq.(16) is
almost same although the effective depth is changed. That is, although energy(W1)increases
slightlywith effectivedepthincrease,the incrementis aboutthree timeswhen the effectivedepth
varies from 16cmto 1600cm,and the effectof effectivedepthon energyis negligible.This result
implies that the analysis is applicableto very large cross sectionsIon the assumptionthat the
energy absorptionis constantand independenton size at the ultimatestate. Figure-16showsthe
distributionof ox an_d0 at failure for differenteffectivedepths. The distributionsare different
fromeffectivedepthand the effect of size appearobviously.
QiBf5g&W
An analysis in which the longitudinalreinforcementratio is variedwas performedfor the RC
cross section shown in Fig.i. The ratiois given nine values from 0.5% to 5.0%. The relationships
between shear force and moment in the failure state are illustratedin Fig.-17.Moment increasein
proportion as the shear force decreaseand maintain constantvalues(momentcapacity)for smaller
shear force. The range in which the moment is constant and the shear force decreases rapidly
correspondsto the flexural failureregion as defined in the analysis (maximumcompressivestrain
of -0.002). On the other hand, the range in which the shear force decreasesgradually corresponds
to the shear failure region. As shownin the figure, the curves for each longitudinalreinforcement
g8
o 1.09(o
e37
A2.0%
E6
^5.0%
V
L
F
V6
U
a 3.09To
Niwa et al.
i
I
0
t5
cG
L
a,(O5
I
/
i/
a)
u)4
.=
u)4
3
+
3
2
+ AIulytical result
l
0.0
0 .0
l.0
2.0
3.0
4.0
5.0
l.0
2.0
3.0
4.0
Moment(tfm)
Fig.-17 Effect of Longitudinal Reinforcement
Table-I Effect of Effecdve Depth
E f e c ti v e
A n a ly sis
A n al y sis
N iw a
D e p th
E q .(16 )
E q .(1 5 )
e t al .
d (c m )
V 1(q
V 2 (tf)
V c (tf)
4 .8 5
5 .2 6
16
4 .8 5
(0 .2 5 )
64
( I.0 )
16 0
(2 5 .0 )
18 .7
14 .8
2 9 .2
4 6 .0
1 8 1.0
4 6 0 .0
S tra in S o ften in g
S train S o fte mi n g
E n e rg y
E n e rgy
W I a (gq c m )
0 .9 2
0 .9 5
(1 .0 )
(2 .5 )
1600
V/V c
(0 .2 6 )
14 .2
5.0
Pt(%)
Fig.-18 Relationbetween LongitudinalReinfTorcement
Rado
and Shear Strength
W 2 a (g f!c m )
0 .8 7 8
0 .8 7 8
(0 .9 1)
(O A 4 )
0 .9 6 3
1 .9 7 6
(1 .0 )
(I .0 )
2 9 .5
0 .9 8
1 .3 19
4 .1 5 7
(1 .3 7 )
(2 .1)
16 6 .2
1.0 9
2 .5 5 5
3 9 .8 7
(2 .6 5 )
(2 0 .l 8 )
(2 .4 6 )
(2 4 .6 )
V1:Analyticalresultusing Eq.(l6) V2:Analyticalresultusing Eq.(15)
Wl:StrainsoftemingenergycorrespondingtoVI
W2:StrainsoftemingenergycorrespondingtoV2
Valuesin parenthesesarenormalizedto d44cm
-102-
ratio are parallel.This impliesthat the effect of longitudinalreinforcementratio on shear strength
is independentof the momentand is a functionof onlythe longitudinalreinforcementratio.
Figure-18 shows the relationshipbetween longitudinalreinforcementratio and shear strengthfor
hu(Vd)-3. The marks '. + f' indicateanalyticalvaluesand the brokenline representsEq.(14).The
difference between the analytical values and Eq.(14) increases with increasing longitudinal
reinforcementratio. The solid line in the figure representsthe shear strengthobtained from the
Z:h%1u-0!n:,nt4h?hI:tCnhilStyhte:cn;dO:r,gs:tsTuo;t5nhatlhs;te:nt$Oee
;rErtgbZieh:?fa6n:;!ynEfe.fi,1.1i
i;Er;?0?d.E;?en:aferotdqH
431
S
insignificant as regards applicability of the analysis, since the difference is less than 10% when
the longitudinalreinforcementratio is in the range of practicaluse(0.3%<Pw<3.0%).
Qjg
The equationsproposedfor shear strengthin the past do not considerthe effects of longitudinal
reinforcementstiffness,sincethe use of steel bars is a premisein the reinforcementof concrete.
Recently,fiber reinforcedplasticO7N),whose stiffnessis lower than that of steel bars, has been
developedas a substitutefor steel bars in the reinforcementof concrete.It has been reported
experimentallythat the shear strengthof concretebeamsreinforcedwithFRP is lowerthan that of
beams reinforcedwith steel bars. Thus,we investigatethe effect of the stiffnessof reinforcement.
tT 2?Salxy
silS.
l6S
kPgeE?cOLm2edf.i?
tETcRhcthcerstsisff3ee3;1.Onf
tshhe.enfonrcFe1;?_n7t
lihVeanieeeuffsom.fO.a2n5a1;slls
a6re
shown in Fig.-19. It is understood that the stiffness of the reinforcementis a factor which
innuences shear strength.The results indicate that the shear strength is proportionalto the 1/4
power of reinforcementstiffness. That is, the effect of longitudinalreinforcementstiffness is
similarto the effectof longitudinalreinforcementratio.
v - vc(Ei /Es)1/4
(18)
Tsuji et al.[8] proposed a method of evaluatingthe shear strength of concretebeams reinforced
with FN by the transformed area of steel bar(As(Eias)) consideringthe differencein stiffness.
The results of analysis prove that this method is adequate. However,the effects of stiffness as
proposed by Tsuji et al. differ from this analysis. The effect of stiffness on shear strength is
representedby a 1/3 power, since Tsuji et al. use the JSCE equation.
The effect of stiffnessis verifiedby a comparisonwiththe experimentaldata presentedby Tsujiet
all. Figure-20showsthe ratio of estimatedand experimentalvalues of shear strength.The
ga)
a
./d-
3
1
O
B6
l.5
0
0
8. l.4
LL
o
L>U i.3
0
L
cG
a)
----__lJt_I__
1
00.25x10
AO.50x10
D1.00x10
^2.00x10
0
0
S]:i
u)I 4
l
2
3
4
o Niwa et al
A rsuji et al.
+Proposed
0.9
I
0.8
A
(
AA
(
rL1
A
A
0.7
0.6
0 .l
5
0 .l5
0.2
0.25
pw(EJEs)
Moment(tfm)
Fig/l9 Effectof sdfFnessof LongitudinalReinfTorcement Fig.-20 Ratio of Estimatedand ExperimentalValues
-103-
Table-2 Effect of CompressiveStrength
f' c
(k g f!c m 2 )
V
O (g f)
f' c /2 8 0
V /4 8 5 0
75
4 15 0
0 .2 6 8
0 .8 5 6
150
4600
0 .5 3 6
0 .9 4 8
280
4850
i .0
1.0
4 50
4950
1 .6 1
i.0 2 1
Table-3 Effect of Tensile Strength
f t
(k g f/cm
I
2)
V
Q g f)
f t /2 8
V /4 8 5 0
l4
3000
0 .5
0 .6 18
28
4 850
1.0
1.0
64 0 0
i.5
1.3 19
42
._
estimated values are obtained from Eq.(14), the method of Tsuji et a1., and Eq.(18). When the
effect of stiffnessis evaluatedby the 1/3 power longitudinalreinforcementstiffnessas proposed
by Tsuji et al., the shear strengthis underestimated.On the other hand, the effect of stiffness is
evaluated more accurately by Eq.(18) where the effect is representedby the 1/4 power of
longitudinalreinforcementstiffness.
a
z%taeknnSi7cS:si!nw%hh:?;ef:thu-?fs8CkO.g/:c:i
s2aln7aeTyEs;secnoagFehp
gas.sva:eiSdntr;n:g3i_eFaoi?:etndta1;:
5tthiaoet
shear strength increasesslightly with increasingf'c, the effect does not appear clear analytically.
The results imply that the compressivestrengthis not an importantfactor in shear strength. The
reason for compressivestrength not having clear effect in the analysis is that initial value of
compressive strength is not analytically important, since the strength varies as a function of
principaltensile strain in the bi-axialstress field of the analysis,
The analysis of variations in tensile strength was performed under the condition of the
compressivestrength of 280kgf/cm2. Three values of tensile strengthwere used: 14, 28 and
42kgf!cm2. The results of analysis are shown in Table-3when M/(Vd)-3. The shear strength
increasesin proportionto tensile strengthand the resultsshowthat the tensile strengthis a major
factor innuencing shear strength.Moreover,it appearsfrom the analysisthat the shear strength
increasesin proportionto approximatelythe 2/3 powerof tensilestrength.
The relationshipbetweencompressiveand tensile strengthis prescribedas the 1/2 powerby the
ACI or the 2/3 power by the JSCE. Therefore,if the above relationsare correct, the result
obtainedfrom the analysisthat the shearstrength
ln tOthe
the ACI
or aoffunctlOnOf
2/3case
power
the.tensile
strength,implythat the shear strengthis a functionlS.fPr(gPc;T1,03ng
(f'c) 4'9 in the JSCE case. We conclude that the effect of concrete strength obtained from the
analysis is almost the same as that givenby Niwa's equation.
5.
CONCLUSIONS
(1) The load carryingcapacity of an RC sectionwas evaluatedby analysisbased on the extended
modified compressionfield theory. Using this analyticalmethod, both shear and flexure failure
strengthcan be evaluatedaccurately.
(2) The solutionalgorithmadopted in this analyticalmethodinvolvesconvergenceof el for each
layer and of ecfor the cross section.This makes possiblefast numericalanalysis in comparison
with Collins'salgorithm,sincethe convergenceparametersare few.
(3) A tension softeningcurve consideringthe effect of size was proposed.Using the proposed
tension softening relationship,the size effect on shear strength can be evaluatedsatisfactorily.It
was proven from a considerationof energy that the analysis is applicableto very large cross
sectionsfor which experimentsare not feasible.
-104-
(4) The effects on the shear strength of RC beams without stirrupsof effectivedepth, longitudinal
reinforcementratio, longitudinalreinforcement stiffness, and concrete strength were investigated
analytically.
(5) The effect of longitudinalreinforcementstiffnesscanbe representedsimilarlyto the effect of
longitudinalreinforcementratio. Analyticalresultsindicatethat the shear strengthis proportional
to the 1/4 power of longitudinalreinforcementstiffness.Moreover,the shear strengthof concrete
beams reinforcedwith FRP can evaluatedaccuratelyby consideringthe effectof the 1/4 powerof
stiffness.
(6) It appears from the analysisthat the shear strength increasesin proportionto the 2/3 power of
tensile strengthand is not innuenced by the compressivestrength.
References
[1] Frank J. Vecchio and Michael P.Collins : The modified compression field theory for
reinforced concreteelementssubjectedto shear, ACI Journal,March-April,pp.219-231,1986.
[2] Frank J. Vecchio and Michael P.Collins : Predictingthe response of reinforcedconcretebeams
subjected to shear using modified compression field theory, ACI Structural Joumal, May-June,
pp.258-268,1988,
[3] Michael P. Collins : Towards a rational theory for RC member in shear, Proc. of ASCE, ST4,
pp.649-666,
April,1978.
[4] Frank J. Vecchio and Michael P.Collins : The response of reinforced concrete to in-plane
shear and normal stresses, Publication No.82-03, Dept. of Civil Eng., Univ. of Toronto, Mar,
1982..
[5] Miyahara T., Kawakami T. and Maekawa K. : Nonlinear behavior of cracked reinforced
concrete plate element under uniaxial compression, Proc. of JSCE, No.378/V-6, pp.249-258, 1987.
[6] Niwa J., Yamada K., Yokozawa K. and Okamura H. : Revaluation of the equation for shear
strength of reinforced concrete beams without web reinforcement, Proc. of JSCE, No.372/V-5,
pp.167-176,
1986.
[7] Okamura H. and Maekawa K. : Non-linear finite element analysis of reinforced concrete, proc.
ofJSCE,
No.360,
pp.1-10,
1985.8.
[8] Tsuji Y., Saito H., Sekijima K. and Miyata S. : Flexura1 and shear behaviors of concrete
beams reinforced with FRP, Proc. ofJCI,
No.10, pp.547-552,
-105-
1988.